Properties

Label 1050.2.i.u
Level $1050$
Weight $2$
Character orbit 1050.i
Analytic conductor $8.384$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,2,Mod(151,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.i (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.21870000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} - 1) q^{4} + q^{6} + (\beta_{4} + \beta_{2}) q^{7} + q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} - 1) q^{4} + q^{6} + (\beta_{4} + \beta_{2}) q^{7} + q^{8} + \beta_{2} q^{9} + (\beta_{5} - \beta_{2} + \beta_1 - 2) q^{11} + \beta_{2} q^{12} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{14} + \beta_{2} q^{16} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{2} - 1) q^{18} + (\beta_{5} - \beta_{4} - \beta_1) q^{19} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{21} + ( - 2 \beta_{5} + \beta_{2} + \beta_1 + 2) q^{22} + ( - \beta_{5} + \beta_{4} + 4 \beta_{2} + \beta_1) q^{23} + ( - \beta_{2} - 1) q^{24} + (\beta_{5} - \beta_{4} - \beta_1) q^{26} + q^{27} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{28} + ( - \beta_{4} + \beta_{3} + 5) q^{29} + ( - \beta_{5} - \beta_{3}) q^{31} + ( - \beta_{2} - 1) q^{32} + (\beta_{5} - 2 \beta_1) q^{33} + (2 \beta_{4} - 2 \beta_{3}) q^{34} + q^{36} + ( - \beta_{5} + \beta_{4} + 4 \beta_{2} + \beta_1) q^{37} + (\beta_{5} + \beta_{3} - \beta_{2} - 1) q^{38} + (\beta_{5} + \beta_{3} - \beta_{2} - 1) q^{39} + (2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{41} + (\beta_{4} + \beta_{2}) q^{42} - 2 q^{43} + (\beta_{5} - 2 \beta_1) q^{44} + ( - \beta_{5} - \beta_{3} - 3 \beta_{2} - 3) q^{46} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} + 3 \beta_1) q^{47} + q^{48} + ( - \beta_{5} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{49} + ( - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{51} + (\beta_{5} + \beta_{3} - \beta_{2} - 1) q^{52} + (2 \beta_{5} + 4 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 5) q^{53} + \beta_{2} q^{54} + (\beta_{4} + \beta_{2}) q^{56} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{57} + (\beta_{4} + 4 \beta_{2} - \beta_1) q^{58} + (2 \beta_{5} + 3 \beta_{3} + 4 \beta_{2} - \beta_1 + 5) q^{59} + ( - 2 \beta_{5} + 4 \beta_{4} + 4 \beta_{2}) q^{61} + (2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{62} + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{63} + q^{64} + (\beta_{5} - \beta_{2} + \beta_1 - 2) q^{66} + ( - 2 \beta_{5} + 2 \beta_{2} - 2 \beta_1 + 4) q^{67} + ( - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{68} + (2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{69} + 6 q^{71} + \beta_{2} q^{72} + ( - 4 \beta_{2} - 4) q^{73} + ( - \beta_{5} - \beta_{3} - 3 \beta_{2} - 3) q^{74} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{76} + ( - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 8 \beta_{2} + \beta_1 + 7) q^{77} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{78} + ( - \beta_{5} + \beta_{4} + 9 \beta_{2} + \beta_1) q^{79} + ( - \beta_{2} - 1) q^{81} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{2} - \beta_1) q^{82} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 6) q^{83} + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{84} - 2 \beta_{2} q^{86} + ( - \beta_{3} - 4 \beta_{2} + \beta_1 - 5) q^{87} + (\beta_{5} - \beta_{2} + \beta_1 - 2) q^{88} + ( - 2 \beta_{5} + 4 \beta_1) q^{89} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - 5 \beta_{2} - \beta_1 - 3) q^{91} + (2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{92} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{93} + ( - 3 \beta_{5} - 3 \beta_{3} + \beta_{2} + 1) q^{94} + \beta_{2} q^{96} + (4 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 7) q^{97} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 4) q^{98} + ( - 2 \beta_{5} + \beta_{2} + \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} + 6 q^{6} + 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} + 6 q^{6} + 6 q^{8} - 3 q^{9} - 3 q^{11} - 3 q^{12} + 6 q^{13} - 3 q^{14} - 3 q^{16} - 6 q^{17} - 3 q^{18} - 3 q^{19} + 3 q^{21} + 6 q^{22} - 9 q^{23} - 3 q^{24} - 3 q^{26} + 6 q^{27} + 3 q^{28} + 24 q^{29} - 3 q^{32} - 3 q^{33} + 12 q^{34} + 6 q^{36} - 9 q^{37} - 3 q^{38} - 3 q^{39} - 18 q^{41} - 12 q^{43} - 3 q^{44} - 9 q^{46} + 3 q^{47} + 6 q^{48} - 12 q^{49} - 6 q^{51} - 3 q^{52} + 9 q^{53} - 3 q^{54} + 6 q^{57} - 12 q^{58} + 12 q^{59} - 6 q^{61} - 3 q^{63} + 6 q^{64} - 3 q^{66} + 6 q^{67} - 6 q^{68} + 18 q^{69} + 36 q^{71} - 3 q^{72} - 12 q^{73} - 9 q^{74} + 6 q^{76} + 3 q^{77} + 6 q^{78} - 24 q^{79} - 3 q^{81} + 9 q^{82} + 24 q^{83} - 3 q^{84} + 6 q^{86} - 12 q^{87} - 3 q^{88} + 6 q^{89} + 18 q^{92} + 3 q^{94} - 3 q^{96} - 48 q^{97} + 21 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 30\nu^{3} + 40\nu^{2} - 70\nu + 13 ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 18\nu^{4} - 15\nu^{3} + 144\nu^{2} + 151\nu - 164 ) / 62 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 13\nu^{4} + 47\nu^{3} - 197\nu^{2} + 461\nu - 133 ) / 62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 13\nu^{4} + 16\nu^{3} - 166\nu^{2} + 151\nu - 102 ) / 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} + 4\beta_{4} - 2\beta_{3} + \beta_{2} - 8\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 16\beta_{5} - 16\beta_{4} + 20\beta_{3} - 9\beta_{2} - 28\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 45\beta_{5} - 60\beta_{4} + 40\beta_{3} - 33\beta_{2} + 55\beta _1 + 139 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
0.500000 + 3.23735i
0.500000 3.05087i
0.500000 + 0.679547i
0.500000 3.23735i
0.500000 + 3.05087i
0.500000 0.679547i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 −2.20942 1.45550i 1.00000 −0.500000 0.866025i 0
151.2 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 0.806615 2.51980i 1.00000 −0.500000 0.866025i 0
151.3 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 1.40280 + 2.24325i 1.00000 −0.500000 0.866025i 0
751.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 1.00000 −2.20942 + 1.45550i 1.00000 −0.500000 + 0.866025i 0
751.2 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 1.00000 0.806615 + 2.51980i 1.00000 −0.500000 + 0.866025i 0
751.3 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 1.00000 1.40280 2.24325i 1.00000 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.i.u 6
5.b even 2 1 1050.2.i.v 6
5.c odd 4 2 210.2.n.b 12
7.c even 3 1 inner 1050.2.i.u 6
7.c even 3 1 7350.2.a.dq 3
7.d odd 6 1 7350.2.a.dp 3
15.e even 4 2 630.2.u.f 12
20.e even 4 2 1680.2.di.c 12
35.f even 4 2 1470.2.n.j 12
35.i odd 6 1 7350.2.a.do 3
35.j even 6 1 1050.2.i.v 6
35.j even 6 1 7350.2.a.dn 3
35.k even 12 2 1470.2.g.h 6
35.k even 12 2 1470.2.n.j 12
35.l odd 12 2 210.2.n.b 12
35.l odd 12 2 1470.2.g.i 6
105.x even 12 2 630.2.u.f 12
140.w even 12 2 1680.2.di.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.b 12 5.c odd 4 2
210.2.n.b 12 35.l odd 12 2
630.2.u.f 12 15.e even 4 2
630.2.u.f 12 105.x even 12 2
1050.2.i.u 6 1.a even 1 1 trivial
1050.2.i.u 6 7.c even 3 1 inner
1050.2.i.v 6 5.b even 2 1
1050.2.i.v 6 35.j even 6 1
1470.2.g.h 6 35.k even 12 2
1470.2.g.i 6 35.l odd 12 2
1470.2.n.j 12 35.f even 4 2
1470.2.n.j 12 35.k even 12 2
1680.2.di.c 12 20.e even 4 2
1680.2.di.c 12 140.w even 12 2
7350.2.a.dn 3 35.j even 6 1
7350.2.a.do 3 35.i odd 6 1
7350.2.a.dp 3 7.d odd 6 1
7350.2.a.dq 3 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{6} + 3T_{11}^{5} + 36T_{11}^{4} + 17T_{11}^{3} + 876T_{11}^{2} + 1323T_{11} + 2401 \) Copy content Toggle raw display
\( T_{13}^{3} - 3T_{13}^{2} - 12T_{13} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{4} + 20 T^{3} + 42 T^{2} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + 36 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$13$ \( (T^{3} - 3 T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + 84 T^{4} + \cdots + 73984 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} + 21 T^{4} + 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$23$ \( T^{6} + 9 T^{5} + 69 T^{4} + 124 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( (T^{3} - 12 T^{2} + 33 T - 24)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 15 T^{4} - 20 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$37$ \( T^{6} + 9 T^{5} + 69 T^{4} + 124 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( (T^{3} + 9 T^{2} - 48 T - 128)^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{6} \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + 141 T^{4} + \cdots + 163216 \) Copy content Toggle raw display
$53$ \( T^{6} - 9 T^{5} + 174 T^{4} + \cdots + 660969 \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + 171 T^{4} + \cdots + 190096 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} + 144 T^{4} + \cdots + 506944 \) Copy content Toggle raw display
$67$ \( T^{6} - 6 T^{5} + 144 T^{4} + \cdots + 153664 \) Copy content Toggle raw display
$71$ \( (T - 6)^{6} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4 T + 16)^{3} \) Copy content Toggle raw display
$79$ \( T^{6} + 24 T^{5} + 399 T^{4} + \cdots + 161604 \) Copy content Toggle raw display
$83$ \( (T^{3} - 12 T^{2} - 27 T + 86)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 6 T^{5} + 144 T^{4} + \cdots + 153664 \) Copy content Toggle raw display
$97$ \( (T^{3} + 24 T^{2} + 117 T + 112)^{2} \) Copy content Toggle raw display
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